Polynomial Regression on Riemannian Manifolds

We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

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“…We now introduce Riemannian polynomials as a generalization of geodesics [15]. Geodesics are generalizations to the …”

Section: Riemannian Polynomialsmentioning

confidence: 99%

“…This work is an extension of the Riemannian polynomial regression framework first presented by Hinkle et al [15]. In Sects.…”

mentioning

confidence: 99%

Intrinsic Polynomials for Regression on Riemannian Manifolds

2014

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“…Finally, the evaluation of β II K requires only weighted averages of two points as it is based on the one-dimensional de Casteljau's algorithm. When simple analytical formulas exist for the Riemannian exponential and logarithm (e.g., for M = S m , see also [22]), this method is very avantageous since the averaging can be based on (13). However, unlike the surfaces of type I and III, the definition of β II K is not symmetric, because it does not satisfy the relation…”

Section: Bézier Surface Definitions Based On Geodesic Averagingmentioning

confidence: 99%

“…This problem has received a fair amount of attention in the literature. Recent contributions can be found in [24,13,30,2].…”

Section: Introductionmentioning

confidence: 99%

Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

Absil¹,

Gousenbourger²,

Striewski³

et al. 2016

SIAM J. Imaging Sci.

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We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve C 0 -and C 1 -continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, C 1 -continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group and two Riemannian shape spaces.Keywords: Composite Bézier surface, Riemannian manifold, differentiability conditions, bending energy.Mathematics Subject Classification (2010): 65D05, 65D07, 53C22, 65K10Figure 1: Differentiable Bézier spline surface on the Riemannian space of shells (Section 6.4) interpolating the red shapes. The gray shapes are points on the Bézier surface driven by the control points in green. Their location indicates where in the R 2 domain they are achieved.

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“…Bayesian nonparametric regression method for Regression on manifolds was proposed in [21,22]. Geodesic regression and Polynomial regression on Riemannian manifolds were proposed in [23] and [24], respectively. A more general case concerning nonparametric regression between Riemannian manifolds (it means that output space R m has dimension m > 1) is studied in [25] where minimization of regularized empirical risk is used for constructing the learned function.…”

Section: Introductionmentioning

confidence: 99%

Statistical Learning on Manifold-Valued Data

Kuleshov

Bernstein

2016

Machine Learning and Data Mining in Pattern Recognition

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Abstract. Regression on manifolds problem is to estimate an unknown smooth function f that maps p-dimensional manifold-valued inputs, whose values lie on unknown Input manifold M of lower dimensionality q < p embedded in an ambient high-dimensional input space R p , to m-dimensional outputs from training sample consisting of given 'input-output' pairs. We consider this problem in which Jacobian J f (X) of function f and Input manifold M should be also estimated. The paper presents a new geometrically motivated method for estimating a triple (f(X), J f (X), M) from given sample. The proposed solution is based on solving a Tangent bundle manifold learning problem for specific unknown Regression manifold embedded in input-output space R p+m and consisting of input-output pairs (X, f(X)), X ∈ M.

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Polynomial Regression on Riemannian Manifolds

Cited by 45 publications

References 22 publications

Intrinsic Polynomials for Regression on Riemannian Manifolds

Intrinsic Polynomials for Regression on Riemannian Manifolds

Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

Statistical Learning on Manifold-Valued Data

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